Unit 1 (Chapter 5): Significant Figures

ErrorEvery measurement has some error in it, based on the device used to get the measurement; some are better than others.

Remember: if you did it right, you guessed a little bit on the last digit.

Both of these are meter sticks, but one is going to be a lot more useful if we want to measure the height of this scientist.

On the other hand, if we're measuring the distance from here to Boston, they're probably about equally good (or equally bad).

Significant FiguresThese are basically rules to let you take a number that somebody gives you, and say “how careful were they?”

1. Non-zeros are always significant. You wouldn't write '1.25 cm' unless you actually measured the tenths and hundredths place.

2. Zeros are where it gets tricky, because there's always the question of “did you measure that place and it really was right on the zero, or did you round?”

We will divide zeros into three types, depending on where in a number they show up...

Significant Figures* Zeros at the beginning of a measurement: placeholders, never significant 0.04 m

(you can see why if you turn it into 4 cm—they go away!)

* Zeros in the middle of a measurement: always significant 4003 m(if you measured the thousands and ones place, presumably

you didn't screw up the hundreds and tens)

* Zeros at the end of a measurement: significant only if there's a decimal point

2300 m (not significant) 230.0 m (significant)(the idea here is that in the first case, you have no way of

knowing if it was rounded to the nearest hundred, but in the second, nobody would write the tenths place unless they measured it)

Some Examples230 g 0.0405 kg 320. miles320 miles 0.0010 mm 3.200 x 105 min

Significant figures underlined

Two notes: 1. for scientific notation, just look at the number; ignore the power of ten.

2. The 320. miles uses the decimal point as a way of saying “hey, I really did measure to the nearest mile, and it just happened to fall on the zero,” because it's at the end and there is a decimal point.

What's the Point?It will help us avoid doing this math:

Or, in more visual terms:

What's the Point?

Here is a red line that is 12 cm long, but with some error in the last digit—it could be 11 cm, or 13.

And here's a blue line that's 5 cm long, but again there's some error—it could be 4, or maybe 6 cm.

Let's do some math with them...

12 cm ± 1cm

5 cm ± 1cm

Adding

If I add them together, my line is now 17 cm long...except that my error has gone up, since the ones place of each was approximated.

It now could be as short as 15.6 cm, or as long as 18.4 cm.

Why not an error of 2 cm? Because it was unlikely that both were a full cm shorter or longer than measured

17 cm ± 1.4 cm

Multiplying

60 cm2 ± 13 cm2

Multiplying gets me an area, but now the smallest possible case is MUCH smaller than the largest. The errors got very large, and it's also not easy to figure out where that 13 came from

Back to Significant FiguresSignificant figures are a set of rules that let us approximate the results shown in the last two slides, but with some advantages:

1. They're quick

2. They don't require any calculus.

3. You just do normal math, and then round your answer.

4. You don't have to find that little ± symbol on your computer.

Adding/SubtractingDo the math as normal, then look for the smallest place that is significant in both numbers, and round to that place:

1050 m + 16.434 m = 1066.434 m → 1070 m

Your answer can't be better than your worst measurement, so you have to round to the nearest tens place.

(this is anti-addition of bromine to an alkene, an organic chemistry concept we will not be covering this year)

Multiplying/DividingDo the math as normal, then look for the smallest number of significant figures, and round to that many in the answer:

1050 m * 16.434 m = 17255.7 m2 → 17300 m2

My worst measurement had three significant figures, so my answer should have three—I keep the ten thousands and thousands place, and round to the nearest hundred.

(math division, not cell division)

Yes, you will sometimes get weird results

100 mL + 0.05 mL = 100 mL ?????

This actually makes sense:

Erlenmeyers are terrible for measuring volume. If the best we can say is “the volume is around 100 mL”, then we have no way of knowing what the effect of and extra 0.05 mL is going to be...

...but it almost certainly isn't exactly 100.05 mL

An exception

How many siblings do you have?

I have 1 brother. There is no error here:I definitely don't have 1.3 brothers, or maybe 0.8 brothers.

(he doesn't look like Mario, either)

It's a counted number, so significant figure rules just don't apply, since they're all about error.

Likewise, unit conversions. 1 km = 1000 m. It is defined to be exactly 1000 m, not 998 m and some rounding error. Again, there is no error in this value.

What Do I Expect?

I expect that you will be within about one significant figure of the correct number. For example:

13.454 cm * 140 cm = 1900 cm2 best answer = 1880 cm2 fine = 2000 cm2 ok, if you must = 1883 cm2 grrrr = 1883.56 cm2 NOOOOOOO!

The last one is basically just lying. It's telling the reader that you have confidence in digits that you cannot possibly know.

What Do I Expect?However, do your rounding after you are done all math, or you get results like:

2 + 2 = 5 (for very large values of 2).

That is:

2.4 + 2.4 = 4.8

Round each to the nearest ones place, and some funny business happens.

SummarySignificant figures express error/uncertainty in a measurement

Non-zeros always count. Zeros count depending on where they are.

They are used when rounding your answer, because your answer can't be better than your worst measurement.

For addition/subtraction, go by place. For multiplication/division, go by how many there are.

Counted numbers and unit conversions don't affect sig figs.

Be reasonable. No numbers like 13.40345893